3. Which of these numbers does not belong to the set of solutions of the inequality 4

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Jerome Terence Hudson
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1 Math Field Day Exam 08 Page. The number is equal to b) c) d) e). Consider the equation 0. The slope of this line is / b) / c) / d) / e) None listed.. Which of these numbers does not belong to the set of solutions of the inequality x x 0? b) c) 0 d) e). Numbers {a, b, c} in this order form an arithmetic progression. Their sum is 7. What is the value of b? 0 b) c) 7 d) e) 9 5. A triangle of sides 6 and 0 and the angle between them 0 is inscribed in a circle. What is the radius of the circle? 7 b) c) 8 d) e) 8 6. The following system of equations x y 6 y 6x has no solutions. b) has exactly one solution. c) has exactly two solutions. d) has exactly four solutions. e) has infinitely many solutions.

2 Math Field Day Exam 08 Page 7. How many a' s exist such that a is between and 50; and the remainders when a is divided by,, and 5 are,,, respectively. 0 b) c) d) e) 8. Suppose. Evaluate.... b) c) d) e) None listed. 9. Compute... 8 b) c) d) e) None listed. 0. Suppose ij k, jk i, ki j, i j k. For any triple a,b, and c, the rule ( ab) c a( bc) is allowed, ji must equal b) 0 c) k d) ij e) None listed.. If AB is the diameter of the circle shown below, and O is its center, what is the angle? A B = 8 b) = 96 c) = d) = 7 e) = 7

3 Math Field Day Exam 08 Page. Let n (so n is the product of the odd integers from to 07). What is the last digit of n? b) c) 5 d) 7 e) 9. How many real numbers x with 0 x 0 are solutions to log 0 ( x ) = sin(x ), where x in sin(x) is in radians? 0 b) c) d) e). Suppose f (x) is a function that satisfies f ( x) 5 f x for all non-zero real numbers x x. What is f ()? 5 b) c) d) 7 8 e) Suppose y is a real number such that y and y y 5 0. What is the closest integer to y? b) 5 c) 6 d) 7 e) 8 6. If a is a positive real number, what is the area of the region in the first quadrant that is bounded above by the graph of y x and below by the graph of y x a? a b) a c) a d) a e) None listed.

4 Math Field Day Exam 08 Page 7. A fair coin is tossed repeatedly. What is the probability that we obtain a total of two tails before we obtain a total of three heads? 6 b) 9 6 c) 5 8 d) 6 e) 8. Find a positive value such that 0 has exactly two distinct real solutions. b) 5 5/8 c) 8 d) 6 /9 e) 9. Which of the following are factors of? I. II. III. I only b) I and II only c) II and III only d) I and III only e) I, II, and III

5 Math Field Day Exam 08 Page 5 0. The line is tangent to a circle of radius centered at and a circle of radius centered at with points of tangency and, respectively. The segment intersects at. If 6, then equals A Q R P B / b) 5/ c) 5/ d) 0/ e) Cannot be determined from the given information.. If and are simultaneous solutions of the system of equations then equals 7 b) 0/ c) /5 d) /5 e) None listed. y x. Given positive real numbers x and y satisfying the equation, what is the value y x y x of? x y b) c) d) e) None listed.

6 Math Field Day Exam 08 Page 6. Find x if x8 where the subscript denotes the base the number is written in. 6 b) 7 c) 75 d) 5 e) None listed.. If n, then what is the value of n? 05 b) 0 c) 0 d) 0 e) Let a, b, c, d and e be numbers such that the average of a and b is 8 and the average of c, d and e is. What is the average of a, b, c, d and e? 6 b) 8/6 c) d) 8/5 e) 0 6. Find all value(s) of x, in [0, π] such that sin x cosx? π/ and 7π/ b) π/ and π/ c) π/ d) Cannot be determined. e) None listed. 7. If a b and a b 7, what is the value of a b? b) 0 c) 7 d) 6 e) 8. If n. what is the value of n in base? b) 0000 c) 0000 d) e) 0000

7 Math Field Day Exam 08 Page 7 9. Suppose that the sum of two numbers is 0 and that the sum of their quotient and its reciprocal is 6. What is the product of the two numbers? b) c) 6 d).5 e) None listed. 0. Solve for x if (x )(x )(x ) (x )(x )(x 5). / b) c) d) / e) /. If the third term of an arithmetic progression is 7 and the seventh term is, what is the tenth term? 0 b) 5 c) 0 d) 6 e) None listed.. Let x and y be two positive numbers such that x y xy. What is the value of x/y? b) c) /5 d) e). sin sin ( 90 ) 0 b) c) Depends upon d) Too little information. e) None listed.. Solve the value of x if x. b).5 c).5 d).75 e) None listed.

8 Math Field Day Exam 08 Page 8 5. What is the perimeter of the orthogonal triangle shown below? 0 a a b) a c) a d) a e) a 6. If the set of three numbers {, 7, x} is supplemented by the number, the arithmetic average of the set will be decreased by. Find number x. In which of these groups does it appear?,, 7 b), 5,8 c),,5 d),,ln e) {0, 0, 00} 7. Sally wants to fence in a rectangular enclosure with the largest possible area. If one side is bounded by a river which does not need a fence and she has material for more feet of fence for the other three sides, what is the length in feet of the diagonal of the rectangular enclosure that she can make with the largest area? b) 5/ c) / d) e) None listed. 8. If A what is the product of b) 0 0 T AA where 0 c) 0 T A is the transpose of A? d) e) None listed.

9 Math Field Day Exam 08 Page 9 9. On a table are two identical boxes. Box A has red and green marbles and box B has red and green marbles. A box was chosen at random and a green marble was drawn. What is the probability the marble came from box A? 5 b) c) 8 d) 5 9 e) None listed. 0. A rectangular box has dimensions 8 by 9 by (all dimensions in inches). What is the length of the longest straight stick which will fit completely inside the box? 7 inches b) inches c) 8 inches d) 5 inches e) None listed F ( x) (( x ) ( x 9) (x 5) (x ) ) is graphed. How many times does the graph of F (x) completely cross the x-axis? 66 b) c) d) 6 e) 0. Suppose log0 x and log0( xy) 6. What is the value of y? 0 7 b) 0 6 c) 07 d) 0 / e) Can t be determined.. Consider triangle ABC with line DE parallel to AC. If the length of BE is x, the length of EC is 5x and the area of the triangle BDE is 6, what is the area of the trapezoid ADEC? B x D E 5x A C 9/ b) 9/60 c) 60/9 d) 90 e) None listed.

10 Math Field Day Exam 08 Page 0. Let p(x) be a cubic polynomial such that p ( ) p() and p() p( / ) 0. What are the roots of p(x)?,, b),, c),, d),, e),, 5. If x = sin 50 then x = cos 50 b) x = cos 0 c) x = cos 0 d) x = tan 0 e) x = tan Evaluate... b) c) d) e) None listed. 7. For any positive integer n, suppose that is a solution to the equation n n n n x x x... x x 0. Evaluate. b) c) d) e) 8. Points A and B are the centers of two circles, tangent to each other at point P. Both circles are tangent to a line and points M and N, respectively. Consider angles MAP and MNP. The angle is equal to A P B M N = 80 b) = 60 c) = d) = 0 + e) = 60

11 Math Field Day Exam 08 Page x 9. Suppose y. Evaluate x. y b) c) d) e) None listed. 50. Suppose f ( x) x. Find a real value of a such that f ( f ( f ( )) 7a. b) c) 0 d) e) None listed. 5. How many solutions satisfying the equation x x x exist? 0 b) c) d) e) None listed. 5. Suppose p and q are distinct positive integers. Which of the following must be a solution of ( x p)( x q) 0? b) c) d) e) None listed. 5. In the given figure, suppose cm, cm, cm, and 5 cm. The ratio of to is B R P Q A C : b) 6:5 c) 5: d) : e) None listed.

12 Math Field Day Exam 08 Page 5. A box contains 6 blue balls, white balls, and red balls. One ball is drawn at random. Its color is noted and the ball is set aside. A second ball is drawn at random. What is the probability that the second ball has the same color as the first? / b) / c) / d) 5/ e) / 55. Find a real number x satisfying simultaneously x y and x y. 0 b) c) d) e) None listed. 56. Assume three real numbers x, y, and z satisfy x y z xy yz zx. What is the maximal value of x y z? b) c) 0 d) e) 57. What is the value of? /50 b) /5 c) /5 d) 8/5 e) /5 58. What is the base representation of /9?. b). c). d). e).5

13 Math Field Day Exam 08 Page 59. Suppose is root of x x 0. What is the value of 6? b) c) 0 d) e) Can t be determined 60. The largest circle that fits inside an equilateral triangle T has area. Each side of T has length b) c) d) e) 6. If the letters A through F are the vertices of a regular hexagon listed in clockwise order, consider the triangle ACE. What is the ratio of the area of the triangle to the area of the hexagon? : b) : c) : d) : e) : 6. What is the lateral surface area of the portion of the cylinder x y 9 that lies on or above the xy-plane and on or below the plane x z 8? 5 b) 8 c) d) e) 7 6. The roots of 0x 6x 9x Ax 0 are x Ax 9x 6x 0 0?,, 5,. What are the roots of,,, 5 b) {,5,,} c) {, 5,,} d),, 5, e) None listed.

14 Math Field Day Exam 08 Page 6. Given the set {,,,, 5, 6}. All of the subsets are written out. The sums of the elements in each subset is computed and then all of these sums are totaled for a final sum. Which of the following best describes this final sum? Less than 500. b) Between 500 and 700. c) Between 700 and 900. d) Between 900 and 000. e) Over Compute b) c) d) e) None listed. 66. Suppose that two real numbers x and y satisfy x y. What is the maximum value xy? b) c) d) e) None listed. 67. What is the smallest integer which DOES NOT solve x x 6 0? b) c) d) e) There is no smallest integer which does not solve this.

High School Math Contest

High School Math Contest University of South Carolina February 4th, 017 Problem 1. If (x y) = 11 and (x + y) = 169, what is xy? (a) 11 (b) 1 (c) 1 (d) 4 (e) 48 Problem. Suppose the function g(x) = f(x)